A theory of risk for two price market equilibria Dilip Madan - - PDF document

A theory of risk for two price market equilibria

Dilip Madan Department of Finance Robert H. Smith School of Business Joint work with Shaun Wang and Phil Heckman

Preview of Results

A theory of risk for two price economies is overlaid

• n an underlying one price economy.

The two price economy is concerned with the failure

• f markets to converge to the law of one price

Equations equations for the two prices are developed with a view to ensuring the acceptability of residual unhedgeable risks in incomplete markets. The acceptability approach results in nonlinear pric- ing operators that are concave for bid prices and con- vex for ask prices. Explicit closed forms for the two prices result when the cone of acceptable risks is modeled using para- metric concave distortions of distribution functions for the residual risk.

With assets marked at bid and liabilities valued at ask prices the theory allows a separation of liability valuation from an associated asset pricing theory. The static two price theory is then extended to its dynamic counterpart by leveraging recent advances made in the theory of non linear expectations and its association with solutions of backward stochastic di¤erence and di¤erential equations. For the hedging of risks we introduce the new crite- rion of capital minimization de…ned as the di¤erence between the ask and bid prices.

Broad View of Two Price Economy

Attention is focused on two prices, the one at which

• ne is guaranteed a purchase or the ask price and

the other the one at which one is guaranteed a sale

• r the bid price.

In e¤ect we contemplate an economy in which most transactions of interest are for products not traded

• n any exchange, for which one may be able to ob-

serve the ask price and or the bid price, but im- portantly there is no possibility of trading in both directions at any observed transaction price. Every transaction is either near or at the ask or near

• r at the bid.

Relevance for Insurance

Products developed for sale to …nal user have both parties holding position to maturity with little if any trading in a secondary market. Products are by design quite speci…c and therefore lack liquidity. Buyers do not buy to sell and sellers do not sell to buy back. Positions are not being reversed and hence there is not much interest in liquidity, but rather in product performance. In the absence of two way transactors it is little won- der that two way prices are absent. What is needed to mark positions is a theory for the two one way prices that do and must prevail in equilibrium.

By focusing attention on a two price economy we model liquidity risk not as an anomaly that is absent in the liquid market but as a core risk especially rele- vant for insurance products even if all …nancial risks are absent.

Dynamic Models for the Two Prices

Insurance contracts typically extend over multiple periods and it is important to analyze the two price economy over multiple periods. The two prices, bid and ask are known to be nonlin- ear and we extend these pricing operators to dynam- ically consistent nonlinear operators by applying the recently developed theory of nonlinear expectations. In this regard we follow Madan and Schoutens (2010) and apply these methods to the pricing of insur- ance claims modeled by increasing compound Pois- son processes.

Hedging in Two Price Economies

The hedging objectives in two price economies turn towards the minimization of ask prices or the maxi- mization of bid prices. Equivalently as suggested in Carr, Madan and Vi- cente Alvarez (2011) one economizes on capital com- mitments measured by the di¤erence between the ask and the bid price. We contrast our capital minimization hedging cri- teria with other classical criteria like variance mini- mization and or the maximization of expected utility. We also apply these new hedging objectives to illus- trate the construction of optimal reinsurance points for contracts insuring losses.

Two Price Economy Pricing Kernels

Consider a two date one period economy trading state contingent claims paying cash ‡ows at time 1 with prices determined at time 0: The claims traded are random variables on a prob- ability space (; F; P) and we suppose that there are some zero cost claims with payouts H 2 H that trade in a liquid market with the same zero cost for trading in both directions. The class of risk neutral measures is then given by R =

n

QjQ P and EQ [H] = 0; all H 2 H

• :

We suppose that an equilibrium has selected a base risk neutral measure Q0 and the set of classically

acceptable risks is then given by the set of positive alpha trades or the set of random variables Ac =

• XjX 2 L1 (; F; P) ; EQ0[X] 0
• :

The de…nition of Ac recognizes that the classical market will accept to buy any amount at a price below the going market price and agree to sell any amount at a price above the price given by the risk neutral expectation. We may de…ne by c the change of measure density c = dQ0 dP and equivalently write that the return RX on X with positive risk neutral price (X) = (1+r)1EQ0[X] > 0 for a periodic interest rate of r; de…ned by RX = X (X) 1

satis…es the condition that EP[RX] r covP (c; RX) ;

• r we have a positive alpha trade or one that earns

in excess of compensation for risk. The point of departure for two price economies from the classical model is the recognition that the half space Ac is too large an acceptance set for realistic economies. For two price economies the acceptance set for the market is de…ned by a smaller convex cone containing the nonnegative random variables. It is shown in Artzner, Delbaen, Eber and Heath (1999) that all such cones are de…ned by requiring a positive expectation under a set of test measures

Q 2 M: The set of risks accepted by the market is then A =

n

XjX 2 L1 ; F; Q0 ; EQ[X] 0; all Q 2 M

• ;

where we suppose that our base measure is Q0 2 M: Madan and Schoutens (2011) determine the set A in equilibrium as the largest set consistent with the aggregate risk held by the market being in a prespec- i…ed small cone containing the nonnegative random variables.

The two prices for a cash ‡ow X of a two price economy are derived from the market’s acceptance cone by requiring that the price less the cash ‡ow for a sale by the market or the other way around for a purchase be market acceptable. Cherny and Madan (2010) show that the unhedged bid and ask prices, with a periodic interest rate of r; b(X); a(X) respectively are given by b(X) = (1 + r)1 inf

Q2M EQ[X]

a(X) = (1 + r)1 sup

Q2M

EQ[X]: Note importantly that the two prices of a two price economy are nonlinear functions on the space of ran- dom variables with the bid price being concave while the ask price is convex by virtue of the in…mum and supremum operations.

The hedging price is determined by maximizing the post hedge bid price or minimizing the post hedge ask price. Formally we have (Cherny and Madan (2010)) that b(X) = sup

H2H

b (X H) a (X) = inf

H2H a(H X):

We now investigate the pricing of risk in our two price economy. We may write the bid and ask prices for X as at- tained at extreme points Qb;X; Qa;X that have den- sities with respect to the base measure Q0 of b;X = dQb;X dQ0 a;X = dQa;X dQ0

and we then have that b(X) = (1 + r)1EP h b;XcX

i

a(X) = (1 + r)1EP h a;XcX

i

If we employ a weighted average as a candidate price de…ning returns e RX relative to this average by

e

RX = X m(X) 1 m(X) = a(X) + (1 )b(X) then we infer the risk pricing equation E[ e RX]r = covP a;X + (1 )b;X c; e RX

• :

Note importantly that by virtue of the nonlinearity

• f the pricing operators of a two price economy the

pricing kernels are no longer independent of the risk being priced.

We build on the classical measure change c of a one price economy an additional illiquidity based measure change given by

• a;X + (1 )b;X

: The second measure change is precisely an illiquidity based measure change as it comes into existence with a bid ask spread associated with an absence of a convergence to a law of one price.

Acceptance Cones Modeled by Concave Distortions

The market primitive of two price economies is the set of zero cost cash ‡ows accepted by the market. This set is a convex cone of random variables con- taining the nonnegative random variables. When the acceptance decision for a random vari- able X is a function solely of its distribution func- tion FX(x) one may evaluate acceptance as shown in Cherny and Madan (2010) by a positive expec- tation under a concave distortion of the distribution function.

Speci…cally for a concave distribution function (u) de…ned on the unit interval and termed the distortion the random variable X is accepted or belongs to the acceptance cone A;just if

Z 1

1 xd (FX(x)) 0:

It is shown in Cherny and Madan (2010) that the set

• f approving measures M are all change of measure

densities on the unit interval Z(u) with

Z 1

1 xZ(FX(x))fX(x)dx 0

for all Z for which L ; where L0 = Z: We mention here two distortions that have been used in earlier work by Cherny and Madan (2010) among

• ther papers and earlier work in the insurance liter-

ature Wang (2000).

These are the transforms minmaxvar; and the Wang transform, . They are de…ned respectively by (u) = 1

• 1 u

1 1+

1+

a(u) = N

• N1(u) + a
• Both these transforms have the desirable property of

derivatives tending to in…nity as u tends to zero and derivatives that tend to zero as u tends to unity.

In terms of distortions one has exact expressions for bid and ask prices (Cherny and Madan (2010)). In this case b(X) = (1 + r)1

Z 1

1 x (FX(x))fX(x)dx

a(X) = (1 + r)1

Z 1

1 y (1 FX(y)) fX(y)dy

So m(X) = (1+r)1EQ0

"

( (FX(X)) +(1 ) (1 FX(X)) X

#

Hence we have that EP[X]r = covP(RX; (FX(X)) +(1 ) (1 FX(X)) c

!

The kernel is then U shaped and we graph the quantile pricing kernel for = :5 and = :5: The kernel cannot be uncorrelated with X as it is a de- terministic function of X:

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5 6 7 8 9 10 kernel for gamma .5 quantile level of pricing kernel

Figure 1: Quantile Risk Pricing Kernel for gamma equal to 0.5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

• 0.2
• 0.15
• 0.1
• 0.05

0.05 0.1 0.15 0.2 Balanced Biased to Bid Biased to Ask

Figure 2: Mid quote base expectation gap at quantiles for digitals For most insurance contracts we have sensitivity in the lower quantiles and so we expect the mid quote to rise above the base expectation as may be ob- served on noting directly that the gap g(a) for a digital at quantile a is g(a) = (a) + 1 (1 a) 2a: We graph in Figure 2 this digital gap.

The gap is positive at quantiles below a half and negative for quantiles above a half.

Dynamic Two Price Economies

We now consider the dynamic valuation of a dis- crete time stochastic claims or receipts process X = (Xt; t = 1; ; T): The valuation is as at time t and is denoted V B

t (X);

V A

t (X) depending on whether we are constructing

a bid price or an ask price. We suppose that the length of the interperiod time interval is h: We suppose the existence of a base risk neutral mea- sure selected by an equilibrium under which one may

construct the risk neutral valuation process V R

t

by V R

t

=

X

jt

B(j) B(t)Xj +EQ0

2 4X

j>t

B(j) B(t)Xj

3 5

=

def

X

jt

B(j) B(t)Xj + W R

t

where B(t) is the time zero discount curve supposed …xed in this exercise. Risk neutral valuation is a well understood linear pric- ing operator and as in the static case it constitutes

• ur starting point.

What we shall present are the nonlinear pricing op- erators for the bid and ask prices. We note in this regard the partitioning of total value into the part of that has been realized and the part

that is yet to be realized by de…ning V A

t (X)

=

def

X

jt

B(j) B(t)Xj +W A

t (X)

V B

t (X)

=

def

X

jt

B(j) B(t)Xj +W B

t (X)

Such nonlinear pricing operators are given by non- linear expectations that are related to solutions of backward stochastic di¤erence equations.

We de…ne risk charges directly for the risk de…ned for example as the zero mean random variable B(t + 1) B(t)

• Xt+1 + W A(t + 1)
• EQ0

"

B(t + 1) B(t)

• Xt+1 + W A(t + 1)

#

: We therefore apply the recursions W A

t (X)

= EQ0

"

B(t + 1) B(t)

• Xt+1 + W A(t + 1)

#

+h sup

Q2M

B @

B(t+1) B(t)

• Xt+1 + W A(t + 1)
• EQ0

B(t+1) B(t)

• Xt+1 + W A(t + 1)
• 1

C A

W B

t (X)

= EQ0

"

B(t + 1) B(t)

• Xt+1 + W B(t + 1)

#

+h inf

Q2M

B @

B(t+1) B(t)

• Xt+1 + W B(t + 1)
• EQ0

B(t+1) B(t)

• Xt+1 + W B(t + 1)
• 1

C A

Drivers for nonlinear expectations based on distortions

The driver for a translation invariant nonlinear ex- pectation is basically a positive risk charge for the ask price and a positive risk shave for a bid price ap- plied to a zero mean risk exposure to be held over an interim. We are then given as input the risk exposure ideally spanned by some martingale di¤erences as ZuMu+1

• r alternatively a zero risk neutral mean random vari-

able X with a distribution function F(x): We consider in the rest of the paper drivers based

• n the distortion minmaxvar: In this case

F B(ZuMu+1) =

Z 1

1 xd(B(x))

F A (ZuMu+1) =

• Z 1

1 xd

1 A(x)

and in particular B(x) = Q0

B B B B @

B(t+1) B(t)

• Xt+1 + W B(t + 1)
• EQ0

B(t+1) B(t)

• Xt+1 + W B(t + 1)
• x

1 C C C C A

A(x) = Q0

B B B B @

B(t+1) B(t)

• Xt+1 + W A(t + 1)
• EQ0

B(t+1) B(t)

• Xt+1 + W A(t + 1)
• x

1 C C C C A

Hedging in Two Price Economies

We note that hedge instruments should have zero means under the base probability measure for oth- erwise these instruments would become vehicles for investment or speculation instead of being used as hedges. With hedges having zero means one may take target cash ‡ows to be hedged to also have a zero mean and hence the hedging criterion should be receptive

• f negative as well as positive cash ‡ows.

A classical criterion often used in studies related to hedging in incomplete markets is variance minimiza- tion or quadratic hedging

This criterion has no parameter with which to re‡ect some degree of aggressiveness or otherwise in hedge design. An often studied alternative criterion is the maxi- mization of expected utility. In the context of two price markets Carr, Madan and Vicente Alvarez (2011) and Madan and Schoutens (2011) suggest capital minimization de…ned as the di¤erence between ask and bid prices. Given that bid and ask prices re‡ect stress parame- ters embedded in distortions capital minimization be- comes a hedging criterion with a parameter allowing for the expression of di¤erent levels of aggressiveness in hedge design.

We present a graph of hedge functions in the con- text of an inhomogeneous Poisson with compound gamma losses. We observe that certainty equivalents are quite asym- metric in their e¤ects on the hedging criterion. Vari- ance as already noted is symmetric but lacks a para- meter.

• 10
• 8
• 6
• 4
• 2

2 4 6 8 10

• 10

10 20 30 40 50 60 Hedge Position in Financial Asset Hedge Criterion Alternative Hedging Criteria for Financial Hedging CE High Risk Aversion High Stress Capital Low Stress Capital Variance CE Low Risk Aversion

Figure 3: Alternative Hedging Criteria for Inhomogeneous Poisson Compound Gamma Losses hedged by security tracking cumulated exceedances

Conclusion

A theory of risk for two price economies is overlaid

• n an underlying one price economy.

The two price economy is concerned with the failure

• f markets to converge to the law of one price and

goes on to develop explicit equations for bid and ask prices with a view to ensuring the acceptability of residual unhedgeable risks in incomplete markets. The acceptability approach results in nonlinear pric- ing operators that are concave for bid prices and con- vex for ask prices. Explicit closed forms for the two prices result when the cone of acceptable risks is modeled using para- metric concave distortions of distribution functions for the residual risk.

With assets marked at bid and liabilities valued at ask prices the theory allows a separation of liability valuation from an associated asset pricing theory. The static two price theory is then extended to its dynamic counterpart by leveraging recent advances made in the theory of non linear expectations and its association with solutions of backward stochastic di¤erence and di¤erential equations. For the hedging of risks we introduce the new crite- rion of capital minimization de…ned as the di¤erence between the ask and bid prices.